A compactness property for prime ideals in Noetherian rings

Abstract

A ring R is compactly packed by prime ideals if whenever an ideal I of R is contained in the union of a family of prime ideals of R, I is actually contained in one of the prime ideals of the family. It is shown that a commutative Noetherian ring is compactly packed if and only if every prime ideal is the radical of a principal ideal. For Dedekind domains this is equivalent to the torsion of the ideal class group and again to the existence of distinguished elements for the essential valuations. If a Noetherian ring R is compactly packed then Krull dim. R 1) be the isolated prime components of the principal ideal (x). Thus \/(x) =PllCP2( .*. . *\Pk, [9, Theorem 10, p. 213]. Since x P, one of the Pi say P1 is necessarily contained in P. We have two cases. Either k =1 in which case P1c,P or k > 1. In that case again P!P2, for from PCP2, we will have P1CPCP2 contradicting the fact that P1, P2 are isolated prime components of (x). Thus in either case, there exists a prime ideal Q. such that P-1IQX and Received by the editors October 13, 1969 and, in revised form, November 22, 1969. AMS Subject Classifications. Primary 1315, 1325; Secondary 1398, 1270.